# Are d8, 10, 12, etc. dice, fair dice?

Given a perfectly formed d8, or d10 or any d dice in Dungeons & Dragons (D&D), are they all fair dice? Is it equally possible to roll any number on any given dice?

I am actually writing a text based, online D&D engine that would allow a DM to create their own world and invite their friends to play that world online. i am writing a cryptographically secure random number generator to roll the dice, but knowing nothing about D&D, i don't know if all the dice are fair.

• You may wish to read the bit on Game Science dice and the tests done by Awesome Dice. – user10595 Sep 6 '14 at 0:00
• Comments are for clarifying the question. Please take tangential discussion to chat. – mxyzplk - SE stop being evil Sep 7 '14 at 17:22
• Why on earth are you using a cryptographically secure RNG for a game? – evilcandybag Sep 7 '14 at 18:39

# Yes, d2, d3, d4, d6, d8, d10, d12, and d20 have uniform distributions

Of these, the d4, d6, d8, d12, and d20 are regular polyhedrons.

The d2 and d10 are not regular polyhedrons, but each face is nonetheless equally-likely.

• I feel like this answer is incomplete; it seems to state an answer without much justification. K. Robin McLean concluded in his 1990 paper "Dungeons, dragons and dice" that the fair dice are the isohedral ones, and the comments at mathoverflow.net/questions/46684 concur. This justifies your answer (aside from d2). ((But I wonder if all that matters is that all the faces are the same and have the same pattern of neighbors, so that the pseudo-deltoidal icositetrahedron is fair. Or maybe a stricter condition is needed and chiral things like pentagonal icositetrahedra are not fair...)) – Mark S. Sep 7 '14 at 5:20
• @MarkS. I agree; it is incomplete. I originally intended to actually justify my statements, rather than assert them, but then realized I did not know how to articulate the mathematics (which are a good deal more involved than I initially assumed) in an understandable way. The paper you've found fits that need very nicely. – KRyan Sep 7 '14 at 14:02
• Well, depending on your definition of "regular", the word is considered ambiguous in maths. The ones you call "regular" are in general called Platonian. All the dice have equal sides, which is a form of regularity as well, and it's the important property here. A small exception is d3 which is a bit more complicated. – yo' Sep 7 '14 at 20:46
• @tohecz Congruent faces is not enough for a die to be fair. The Gyroelongated square bipyramid is certainly not a fair die shape. – Mark S. Sep 9 '14 at 1:07
• @MarkS. Ah, again confusion. I should have written equivalent, not equal. – yo' Sep 9 '14 at 6:03

Perfectly formed is a bit tricky, but yes.

If a solid is rotationally symmetric such that one side being face up is the equivalent of any other side being face up, then there are no differences, and as such have the same probability.

If you're interested in testing the fairness of real life dice, I'd suggest looking here: https://stats.stackexchange.com/questions/3194/how-can-i-test-the-fairness-of-a-d20

• I would describe that as face down, as I have never seen a d4 land in a way I would describe as having an up face, but they are still fair. – Joel Harmon Sep 5 '16 at 1:28

It is very easy to see if a die is fair, just see if you can deduce anything about the outcome without the faces. For example, a coin without faces is just a disc and a d6 without faces is just a cube. You can't tell anything from a resting cube or disc, so they are both fair. The same goes for all dice commonly used in rpg games. Technically a coin can stand on its edge since it is not a perfect disc, but if we ignore those cases it is fair. In the same way we can create all kinds of dice by just rolling a pen with the amount of sides we want. For example we can get a fair d5 by rolling a pentagonal pen.

Fairness of a die depends a lot on fine details (are the edges rounded exactly the same? is the die from a homogenous material? this kind of thing), but from a theoretical, purely geometric point of view, all classical dice are "fair", in the following sense: each face of the die could be sent to any other without deforming the die, and keeping the contour of the die globally unchanged. This is enough symmetry to ensure that no face is favored over any other. For the mathematically inclined: the group of isometries of each solid acts transitively on the faces.

The classical Platonician volumes (D4, D6, D8, D12, D20) have even more symmetry than that, but even the usual D10 has enough symmetry for that.

A die will have equal likelihood of each outcome if (a) each face has the same surface area and (b) the center of gravity of the actual die aligns with the center of the polyhedron. The reason dice have a set pattern for number faces (for example, in a D20 1 is opposite 20, and neighbor to 19, which is opposite 2, which in turn is neighbor to 18) is that if the center of gravity is misaligned due to imperfections in the manufacturing process, the arithmetic mean of the outcomes over a large number of throws will still aproach (1+number of faces)/2.

• There are more things that can make a die unfair. For instance if the surfaces were not equally smooth, that is, if the have different friction rates. – Flamma Sep 7 '14 at 18:05

If you're writing a program to generate uniformly random numbers, and you do it correctly, every number is equally likely, whether or not the physical dice are equally likely to roll a given result is moot.

You could create a number generator with a quadrillion possible results and provided your coding was on point and your algorithm for generating uniformly random numbers was correct, each face would be equally likely to occur.

The randomness of dice is in part due to their shape, but someone skilled at throwing dice a specific way could still end up with a much more favorable result than someone throwing them randomly.

Dominic LoRiggio was a man who went to a casino and played Craps with d6's, but threw the dice in such a way that the dice stuck together when thrown, and barely tapped against the far wall of the craps table, which let him control which number the dice would end up on.

There are also flaws in the dice that are made using specific manufacturing processes that create inherent flaws in the center of gravity. More about this method and how it can be detected can be found here.

You can also use a cylinder with equally wide squared off triangular facets to make cylindrical dice with any amount of numbers on them that will produce a random result when thrown. These dice look like this.

As KRyan said, most of the dice are in fact, balanced and when thrown will land on a random result provided their manufacturing method doesn't create any deviance in its center of gravity.. but if you're going to be coding a random number generator, that doesn't matter in the slightest.

• I think this misunderstands the question. If a (e.g.) d12 is an inherently fair die, then it can be modelled with a virtual roller that gives all possible results equally often. If it's not a fair die though, then to accurately model D&D probabilities, the virtual roller has to be written to account for uneven result odds. Whether classic D&D dice give results in a flat distribution is then important to know before writing virtual roller software. – SevenSidedDie Sep 5 '16 at 23:17
• You wouldn't be able to determine whether or not the dice is fair unless you tested its center of gravity, even if it is a polyhedron with uniform distributions. There can be perfect dice that are equally likely, but there are also flawed dice that favor specific results when tested. If a die that normally has a uniform distribution has a center of gravity that is off it will favor one side regardless of how many times its rolled, so the chances are likely that the players at the table could be using either fair or unfair dice. It's not impossible to model this, but it would be difficult. – Sandwich Sep 5 '16 at 23:23
• Yes, that's still the misunderstanding that I'm getting at. That comment and this is answer is assuming that everyone already knows that these shapes are supposed to give (approximately) uniform distributions. The question asker doesn't (didn't) know that. – SevenSidedDie Sep 5 '16 at 23:26
• This answer is challenging the frame of the question. Whether or not they are fair or unfair is a moot point because a dices fairness isn't a variable that can be measured accurately 100% of the time. – Sandwich Sep 5 '16 at 23:28
• Yes, I grasped that. But the challenge is premised on the misunderstanding that I have attempted, failed, and now resign myself to not successfully explaining. – SevenSidedDie Sep 5 '16 at 23:31